Books are a symbol of the world that existed before the setting in the story; they contain what is truly forbidden knowledge in the world that Montag lives in.
Fire is something that is used for "cleansing" in the sense that it burns books, people, and houses that are associated with books in any way.
The mechanical hound searches for people who are running away from any punishement they are due; later, the hound is searching for Montag who is able to change his body chemistry and escape the hound.
The parlor walls are the place where the tv screens that Mildred and her friends are so attached to are mounted. They represent the family for Mildred and the people around her.
Mirrors represent the way in which people should look closely at themselves and see what they are like all the way inside.
The Phoenix is a mythological symbol which is periodically burned to ashes and then rises from the ashes more and more beautiful.
The last has two references: first, Montag remembers something from his childhood when he tried to keep sand in a sieve and the more he loaded into it, the more fell through. As an adult trying to memorize the Bible, the more he puts in his memory, the more that seems to fall through. He is very frustrated.